Optimal strategies for equal-sum dice games

Abstract

In this paper, we consider a non-cooperative two-person zero-sum matrix game, called dice game. In an ( n , σ ) dice game, two players can independently choose a dice from a collection of hypothetical dice having n faces and with a total of σ eyes distributed over these faces. They independently roll their dice and the player showing the highest number of eyes wins (in case of a tie, none of the players wins). The problem at hand in this paper is the characterization of all optimal strategies for these games. More precisely, we determine the ( n , σ ) dice games for which optimal strategies exist and derive for these games the number of optimal strategies as well as their explicit form.